In a previous presentation the two choices for the generalized Runge vector were given as

R={RL[pcosαr-rcosφp]RL[−psinαr+rsinφp]

where the φ measures the angle between the radius vector and a fixed direction, while α measures the angle between the linear momentum vector and the same fixed direction. This short presentation will exhibit explicitly the constancy of these two vectors for power potentials with arbitrary real exponents.

If the angle between the radius and linear momentum vectors is designated by γ, then as pointed out at the end of the previous presentation there is a simple geometric relationship
α = φ + γ among the three angles. The angle γ can be evaluated quite easily as

γ=cos−1[(r·p)rp]

using dynamic quantities available from a numerical integration of the equations of motion. The results of the numerical integration will be shown as interactive graphics.

Since the overall magnitude of the two vectors is not important in the context of this discussion, the vectors

pcos(φ+γ)r-rcosφp−psin(φ+γ)r+rsinφp

will be evaluated numerically and displayed for bound states. The vectors are constant as well for states that are not bound, but display of these states is less convenient because the evolution is not resticted in radial extent.

The Hamiltonian for the evolution of dynamic quantities is

H=p22m+ark=pr22m+L22mr2+ark=pr22m+Veffective=E

with coupling constant a and arbitrary real exponent k on the power potential. The system can only have bound states when the effective potential, consisting of the power potential plus the angular momentum term, has a concavity or potential well. The state is bound when the value of the energy energy touches the potential well on both sides.

The conditions for bound states are exhibited in the following interactive graphic. The mass parameter has been taken as unity, which is equivalent to absorbing it into the energy and coupling constant. The light blue line is the effective potential and the green line is the value of the energy:

Manipulation of the parameters in the graphic indicates that there are bound states for all positive exponent values as long as the coupling constant and energy are also positive. It also shows that there are bounds states for negative exponents between 0 and −2 as long as the coupling constant and energy are also negative. There are bounds states of negative energy possible for exponents more negative than −2, but then the effective potential well includes the origin and orbits for these states would need to pass through the center of interaction for the problem: these states will not be considered in what follows.

In summary, positive energy states have positive exponent and coupling constant, and negative energy states have negative exponent and coupling constant with a lower bound on the exponent. The two cases will be presented separately for convenience.

Numerical integration of the equations of motion is simplest in the Cartesian coordinates of the two-dimensional plane of orbit. With the usual designations

r2=x2+y2p2=px2+py2

for the two pairs of conjugate variables in the plane, the equations of motion are

x·=pxmp·x=−kax(x2+y2)k2-1y·=pymp·y=−kay(x2+y2)k2-1

The remaining dynamic quantity is the angle between the radius and Runge vectors, which can be evaluated from the statement of conservation of angular momentum:

φ·=Lmr2=Lm(x2+y2)

It is now straightforward to integrate this set of five coupled differential equations using a simple Euler method. The mass parameter will again be taken as unity, and the evolution will begin with the conditions r = [ 1 , 0 ] and p = [ 0 , L ].

From a practical point of view one does not need to perform any integration for the angular variable, since the given intial conditions are equivalent to

φ=cos−1[xx2+y2]=sin−1[yx2+y2]

but this approach has the sense of presupposing the final vector will be constant without allowing the parts to come together a bit independently. One could also explicitly integrate the temporal derivative
α·
of the angle between the linear momentum and Runge vectors, but given the simple relationship among dynamic angles this would merely add processing overhead with no gain.

The interactive graphics have restrictions placed on input parameters in order to reduce the possibility of significant error in the numerical integration. What is critical is to allow enough variation around known soluble cases to see that variation of these parameters still leads to resultant constant vectors. For negative energy states the variation occurs around the Kepler solution, which is the historical source for the Runge vector. For positive energy states the variation occurs around the solution for the simple harmonic oscillator, a case that is fully soluble because of its well-known mathematical relationship to the Kepler problem.

The two choices of the invariant Runge vector are displayed in red and blue, with their constituent portions in lighter shades of each color. The radius vector is shown in black, with the momentum vector in gray extending from its tip as is appropriate for an orbiting body. Two periods of the orbit appear in light gray to emphasize that the constant vectors exist even for nonclosed orbits.

The interactive graphic for bound states of negative energy is

and the interactive graphic for bound states of positive energy is

In both graphics one can see a portion of each choice for Runge vector lying along the radius vector, and a second portion extending from the first that is parallel to the momentum vector. The vector sum of the two portions is in all cases a constant vector for the motion.

The results of this presentation are in a definite sense trivial, since it was pointed out in another presentation that the Runge vector is essentially a rearrangement of the components of the angular momentum tensor. The two vectors displayed are really

[0,−L]and[−L,0]

which are of course constant. The fun in this little presentation is evaluating the two complicated parts and adding them together for a trivial result. It should also be stressed that doing this shows that the Runge vector is a constant for all spherically symmetric potentials, in this case arbitrary powers of the radial separation, and is not restricted to the Kepler potential or the simple harmonic oscillator.